Uniform Spanning Forest Research Articles

Watermelons on the half-plane

We study the watermelon probabilities in the uniform spanning forests on the two-dimensional semi-infinite square lattice near either the open or closed boundary to which the forests can or cannot be rooted, respectively. We derive universal power laws describing the asymptotic decay of these probabilities with the distance between the reference points growing to infinity, as well as their non-universal constant prefactors. The obtained exponents match with the previous predictions made for the related dense polymer models using the Coulomb gas technique and conformal field theory, as well as with the lattice calculations made by other authors in different settings. We also discuss the logarithmic corrections some authors argued to appear in the watermelon correlation functions on the infinite lattice. We show that the full account for diverging terms of the lattice Green function, which ensures the correct probability normalization, provides the pure power law decay in the case of semi-infinite lattice with the closed boundary studied here, as well as in the case of infinite lattice discussed elsewhere. The solution is based on the all-minors generalization of the Kirchhoff matrix tree theorem, the image method and the developed asymptotic expansion of the Kirchhoff determinants.

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely. This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (Probability on Trees and Networks (2016) Cambridge Univ. Press) in the negative. A version of our argument gives an example of a nonunimodular transitive graph where WUSF≠FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang (Electron. J. Probab. 26 (2021) Paper No. 141).

Connectedness of the Free Uniform Spanning Forest as a function of edge weights

Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in [4] that the Free Uniform Spanning Forest (FSF) of this graph may not be connected, but the dependence of this connectedness on H remains somewhat mysterious. We study the case when a positive weight w is put on the edges of the H-copies in G, and conjecture that the connectedness of the FSF exhibits a phase transition. For large enough w we show that the FSF is connected, while for a wide family of H and T, the FSF is disconnected when w is small (relying on [4]). Finally, we prove that when H is the graph of one edge, then for any w, the FSF is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.

Uniform spanning forests on biased Euclidean lattices

La forêt couvrante uniforme (notée USF) sur un graphe connexe, infini et localement fini avec conductance c(·), est définie comme la loi limite de l’arbre couvrant uniforme de sous-graphes finis. Suivant le graphe et les conductances, la mesure USF ne porte pas nécessairement sur l’ensemble des arbres couvrants. Pemantle (Ann. Probab. 19 (1991) 1559–1574) a démontré que sur Zd muni de la conductance unité c=1, USF porte sur les arbres couvrants si et seulement si d≤4. Dans ce travail, nous étudions USF associée aux conductances c(e)=λ−|e|, où |e| est la distance entre l’arête e et l’origine, et λ∈]0,1[ est un paramètre fixé. Notre résultat principal montre que dans ce cas, USF porte sur un nombre fini d’arbres si et seulement si d=2 ou 3. Plus précisément, nous prouvons que la forêt couvrante uniforme contient 2d arbres si d=2 ou 3, et contient une infinité d’arbres si d≥4. Notre approche s’appuie sur une analyse du rayon spectral et de la vitesse de la marche aléatoire λ-biaisée sur Zd.

Infinite-step stationarity of rotor walk and the wired spanning forest

We study rotor walk, a deterministic counterpart of the simple random walk, on infinite transient graphs. We show that the final rotor configuration of the rotor walk follows the law of the wired uniform spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF. This result holds for all graphs for which each tree in the wired spanning forest has one single end almost surely. This answers a question posed in a previous work of the author (Chan 2018).

Weights of uniform spanning forests on nonunimodular transitive graphs

Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest. Regarding the free uniform spanning forest, we consider several families of nonunimodular transitive graphs. We show that the free uniform spanning forest is the same as the wired one on Diestel--Leader graphs. For grandparent graphs, we show that the free uniform spanning forest is connected and has branching number bigger than one. We also show that each tree of the free uniform spanning forest is heavy and has branching number bigger than one on a free product of a nonunimodular transitive graph with one edge when the free uniform spanning forest is not the same as the wired.

Induced graphs of uniform spanning forests

Étant donné un sous-graphe $H$ d’un graphe $G$, le graphe induit de $H$ est le plus grand sous-graphe de $G$ dont l’ensemble de sommets est le même que celui de $H$. Notre article concerne les graphes induits des composants connexes de $\mathsf{WSF}(G)$, la forêt recouvrante uniforme câblée sur $G$, et, dans une moindre mesure, $\mathsf{FSF}(G)$, la forêt recouvrante uniforme libre. Nous montrons que le graphe induit de chaque composant de $\mathsf{WSF}(\mathbb{Z}^{d}$) est presque sûrement récurrent lorsque $d\ge 8$. De plus, la résistance effective entre deux points du rayon de l’arbre à l’infini au sein d’un composant croît linéairement lorsque $d\ge 9$. Pour tout graphe transitif à sommets $G$, nous établissons la propriété de rééchantillonnage suivante: Étant donné un sommet $o$ dans $G$, soit $\mathcal{T}_{o}$ le composant de $\mathsf{WSF}(G)$ qui contient $o$ et $\overline{\mathcal{T}_{o}}$ son graphe induit. Conditionné sur $\overline{\mathcal{T}_{o}}$, l’arbre $\mathcal{T}_{o}$ est distribué comme $\mathsf{WSF}(\overline{\mathcal{T}_{o}})$. Pour tout graphe $G$, nous montrons également que si $\mathcal{T}_{o}$ est le composant de $\mathsf{FSF}(G)$ qui contient $o$ et $\overline{\mathcal{T}_{o}}$ est son graphe induit, alors conditionné sur $\overline{\mathcal{T}_{o}}$, l’arbre $\mathcal{T}_{o}$ est distribué comme $\mathsf{FSF}(\overline{\mathcal{T}_{o}})$.

Indistinguishability of collections of trees in the uniform spanning forest

Dans cet article, nous prouvons le théorème d’indistinguabilité suivant pour les $k$-uplets d’arbres dans le modèle de forêt couvrante uniforme sur $\mathbb{Z}^{d}$ : supposons que $\mathscr{A}$ est une propriété d’un $k$-uplet de composantes connexes qui est stable par modification finie de la forêt, alors ou bien chaque $k$-uplet satisfait la propriété $\mathscr{A}$ presque sûrement, ou bien aucun ne la satisfait presque sûrement. Ce résultat étend le théorème d’indistinguabilité de l’auteur et de Nachmias (2016) qui ne couvrait que le cas d’arbres pris individuellement. Notre résultat s’applique plus généralement à tout graphe vérifiant la propriété de Liouville et pour lequel toutes les composantes connexes de la FCU ont un seul bout.

Mean-field avalanche size exponent for sandpiles on Galton\u2013Watson trees

We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as t^{-1/2}. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on {mathbb {Z}^d}, dge 3, and other transient graphs.

Universality of high-dimensional spanning forests and sandpiles

We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of mathbb {Z}^d, dge 5. In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph. In the case of mathbb {Z}^d, dge 5, some of our results regarding critical exponents recover earlier results of Bhupatiraju et al. (Electron J Probab 22(85):51, 2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work.

Geometry of Uniform Spanning Forest Components in High Dimensions

AbstractWe study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^{d}$ and give bounds on the size of balls in the intrinsic metric.

UNIFORM SPANNING FORESTS OF PLANAR GRAPHS

We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.

Rotor Walks on Transient Graphs and the Wired Spanning Forest

We study rotor walks on transient graphs with initial rotor configuration sampled from the oriented wired uniform spanning forest (OWUSF) measure. We show that the expected number of visits to any vertex by the rotor walk is at most equal to the expected number of visits by the simple random walk. In particular, this implies that this walk is transient. When these two numbers coincide, we show that the rotor configuration at the end of the process also has the law of OWUSF. Furthermore, if the graph is vertex-transitive, we show that the average number of visits by $n$ consecutive rotor walks converges to the Green's function of the simple random walk as $n$ tends to infinity. This answers a question posed by Florescu et al. (2014).

The component graph of the uniform spanning forest: transitions in dimensions 9,10,11,ldots

We prove that the uniform spanning forests of mathbb {Z}^d and mathbb {Z}^{ell } have qualitatively different connectivity properties whenever ell >d ge 4. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8. To separate dimensions 5, 6, 7, and 8, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph. Our result sharpens a theorem of Benjamini, Kesten, Peres, and Schramm, who proved that the diameter of the component graph increases by one every time the dimension increases by four.

Indistinguishability of the components of random spanning forests

We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres and Schramm for Cayley graphs, which have been open up to now. Our methods apply to a more general class of percolations, those satisfying “weak insertion tolerance”, and work beyond Cayley graphs, in the more general setting of unimodular random graphs.

Interlacements and the wired uniform spanning forest

We extend the Aldous–Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman’s random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of ‘excessive ends’ in the WUSF is nonrandom in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm [Electron. J. Probab. 13 (2008) 1702–1725], while the third extends a recent result of the author. Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.

Wired cycle-breaking dynamics for uniform spanning forests

We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSF is one-ended almost surely in every supercritical Galton-Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm. Our proof introduces and exploits a family of Markov chains under which the oriented WUSF is stationary, which we call the wired cycle-breaking dynamics.

Indistinguishability of trees in uniform spanning forests

We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF) of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm. We use this to answer positively two additional questions of Benjamini, Lyons, Peres and Schramm under the assumption of unimodularity. We prove that on any unimodular random rooted network, the FUSF is either connected or has infinitely many connected components almost surely, and, if the FUSF and WUSF are distinct, then every component of the FUSF is transient and infinitely-ended almost surely. All of these results are new even for Cayley graphs.

Solvable and algebraic systems on infinite ladder

We consider two solvable models with equal entropy on the infinite ladder graph Z×{1,2}: the uniform spanning forest (USF), the abelian sandpile (ASM). We show that the symbolic models (abelian sandpile and spanning forest) are equal entropy symbolic covers of a certain algebraic dynamical system. In the past results of this nature have been established for sandpile models on lattices Zd. But we present a first example in case of spanning trees.

Invariant coupling of determinantal measures on sofic groups

To any positive contraction $Q$ on $\ell ^{2}(W)$, there is associated a determinantal probability measure $\mathbf{P}^{Q}$ on $2^{W}$, where $W$ is a denumerable set. Let ${\rm\Gamma}$ be a countable sofic finitely generated group and $G=({\rm\Gamma},\mathsf{E})$ be a Cayley graph of ${\rm\Gamma}$. We show that if $Q_{1}$ and $Q_{2}$ are two ${\rm\Gamma}$-equivariant positive contractions on $\ell ^{2}({\rm\Gamma})$ or on $\ell ^{2}(\mathsf{E})$ with $Q_{1}\leq Q_{2}$, then there exists a ${\rm\Gamma}$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination $\mathbf{P}^{Q_{1}}\preccurlyeq \mathbf{P}^{Q_{2}}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when ${\rm\Gamma}$ is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures $\mathbf{P}^{Q}$ as above are $\bar{d}$-limits of finitely dependent processes. Thus, when ${\rm\Gamma}$ is amenable, $\mathbf{P}^{Q}$ is isomorphic to a Bernoulli shift, which was known before only when ${\rm\Gamma}$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.